Related papers: Invariant random subgroups and action versus repre…

The concept of (stable) weak containment for measure-preserving actions of a countable group $\Gamma$ is analogous to the classical notion of (stable) weak containment of unitary representations. If $\Gamma$ is amenable then the Rokhlin…

Let $G$ be a closed permutation group on a countably infinite set $\Omega$, which acts transitively but not highly transitively. If $G$ is oligomorphic, has no algebraicity and weakly eliminates imaginaries, we prove that any probability…

Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p.…

We consider two numerical entropy--type invariants for actions of $\Zk$, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case,…

We introduce a natural pseudometric on the space of actions of d-generated groups. In this pseudometric, the zero classes correspond to the weak equivalence classes defined by Kechris, and the metric identification is compact. We achieve…

We show that countable non-abelian free groups admit uncountably many mutually singular elementwise conservative non-singular random subgroups, which are supported on infinite subgroups of infinite index and singular with respect to every…

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic…

Namely, we prove that if $G$ is a countable, discrete group and $f\in M_{n}(\Z(G))$ is invertible on $\ell^{2}(G)^{\oplus n},$ but $f$ is not invertible in $M_{n}(\Z(G))$, then the measure-preserving action of $G$ on $X_{f}$ equipped with…

We prove that for any free ergodic probability measure preserving action \F_n \actson (X,\mu) of a free group on n generators \F_n, 2 \leq n \leq \infty, the associated group measure space II_1 factor $L^\infty(X) \rtimes \F_n$ has…

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the…

Let $G$ be a second-countable, locally compact Hausdorff groupoid equipped with a Haar system. This paper investigates the weak containment of continuous unitary representations of groupoids. We show that both induction and inner tensor…

Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman…

The $f$-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.

We prove a pointwise ergodic theorem and a maximal inequality for actions of amenable groups on noncommutative measure spaces. To do so, we establish a square function estimate quantifying the difference between ergodic averages and some…

Ergodic and combinatorial results obtained in [10] involved measure preserving actions of the affine group ${\mathcal A}_K$ of a countable field $K$. In this paper we develop a new approach based on ultrafilter limits which allows one to…

Given a transformation $T$ of a standard measure space $(X,\mu)$, let $\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,\mu)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in this…

In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative $L^1$-spaces. As a further consequence, in…

We exhibit rationally ergodic, weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.

We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (\`a la Connes-Feldman-Weiss) precisely when it is 'hypercompact'. This implies an uncountable version of the Ornstein-Weiss Theorem and…

It is shown that there exist a subsequence for which the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space converge almost everywhere provided that the maps are weakly…